Sample 9: This sample has: a. no cleavage (it fractures) b. 1 cleavage plane c. 2 cleavage plane

Sample 9: This sample has: a. no cleavage (it fractures) b....

Key Concepts

Mineral Cleavage

Mineral cleavage is the tendency of a mineral to break along flat, parallel planes of weakness in its crystal lattice. This characteristic is defined by the alignment of atoms within the mineral and can be a critical property used for identification, as it reflects the internal structure and symmetry of the crystal.

Cleavage Planes

Cleavage planes are the defined, smooth surfaces along which a mineral splits when subjected to stress. The number and orientation of these planes (including the angles between them) provide important clues about the mineral’s crystallographic structure and are used to differentiate between different minerals.

Mineral Fracture

Mineral fracture occurs when a mineral breaks in an irregular manner that does not follow any plane of weakness due to the absence of cleavage. This produces surfaces that are often rough, curved, or jagged, and it is an equally significant property for mineral identification.

Transcript

00:03 We're told that the fracture angle under pure compression of a randomly selected specimen of fiber -reinforced polymer matrix composite material is normally distributed with a mean value of 53 and a standard deviation of one.

00:24 In part a, we're told that a random sample of four specimens is selected, and we're asked to find the probability that the sample mean fracture angle is at most 54.

00:36 Well, we'll let x bar note the sample mean fracture angle of the four samples.

01:14 Now the individual fracture angles are normally distributed, so it follows that x bar is also a normal random variable.

01:26 In particular, with a mean, the expected value of x bar is the same as the mean mu, which we're given is 53, but with the standard deviation, standard deviation of x bar, well this is going to be the standard deviation of the population sigma over square root of the number of samples n.

01:58 We're given that sigma is one and that n was what we had four samples so root 4.

02:06 So this is one half or 0 .5.

02:15 And therefore it follows that probability that this is less than or equal to 54, well, this is the same as the probability that the standard normal vector, random variable z, is less than or equal to 54 minus the mean 53 over the standard deviation, 0 .5, which is the same as probability that z is less than or equal to 2, which is the same as 5 of 2.

03:02 And using a table or a computer, we calculate that this is 0 .972 approximately.

03:19 We're also asked to find the probability that the sample mean fracture angle lies between 53 and 54.

03:29 So the probability that x bar lies between 53 and 54.

03:36 Well, this is the probability that the standard normal random variable z lies between 53 minus the mean 53 over the standard deviation 0 .5 and 54 minus the mean 53 over the standard deviation 0 .5, which is the same as the probability that z lies between 0 .0 and 2.

04:13 Which of course is the same as phi of two minus five of zero...

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